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Properties

Label 2-546-273.173-c1-0-6
Degree $2$
Conductor $546$
Sign $0.630 - 0.776i$
Analytic cond. $4.35983$
Root an. cond. $2.08802$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−1.72 + 0.0896i)3-s + (−0.499 + 0.866i)4-s + (3.27 + 1.89i)5-s + (0.942 + 1.45i)6-s + (2.49 + 0.889i)7-s + 0.999·8-s + (2.98 − 0.310i)9-s − 3.78i·10-s − 5.80·11-s + (0.787 − 1.54i)12-s + (0.879 + 3.49i)13-s + (−0.475 − 2.60i)14-s + (−5.84 − 2.98i)15-s + (−0.5 − 0.866i)16-s + (−2.26 + 3.93i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.998 + 0.0517i)3-s + (−0.249 + 0.433i)4-s + (1.46 + 0.846i)5-s + (0.384 + 0.593i)6-s + (0.941 + 0.336i)7-s + 0.353·8-s + (0.994 − 0.103i)9-s − 1.19i·10-s − 1.75·11-s + (0.227 − 0.445i)12-s + (0.243 + 0.969i)13-s + (−0.126 − 0.695i)14-s + (−1.50 − 0.769i)15-s + (−0.125 − 0.216i)16-s + (−0.550 + 0.953i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 - 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.630 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.630 - 0.776i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.630 - 0.776i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.928128 + 0.441718i\)
\(L(\frac12)\) \(\approx\) \(0.928128 + 0.441718i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (1.72 - 0.0896i)T \)
7 \( 1 + (-2.49 - 0.889i)T \)
13 \( 1 + (-0.879 - 3.49i)T \)
good5 \( 1 + (-3.27 - 1.89i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + 5.80T + 11T^{2} \)
17 \( 1 + (2.26 - 3.93i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 2.85T + 19T^{2} \)
23 \( 1 + (-4.19 + 2.42i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.20 + 1.27i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.824 - 1.42i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.865 - 0.499i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.48 - 2.01i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.445 + 0.772i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.23 - 3.59i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.41 - 3.12i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-8.67 - 5.00i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 3.64iT - 61T^{2} \)
67 \( 1 + 10.6iT - 67T^{2} \)
71 \( 1 + (-6.77 - 11.7i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.19 + 3.79i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.40 - 14.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.9iT - 83T^{2} \)
89 \( 1 + (-1.89 + 1.09i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.30 + 10.9i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.76907057692045171677498276046, −10.44195116731555210216604778865, −9.456114728179908094180348229195, −8.429747973761713941152889154830, −7.21683410764723967807699690941, −6.23158485898438440092015870862, −5.40894263059318792577259465387, −4.44157763605951651685622232046, −2.54629989705798290924837870633, −1.73291061045237025707702522264, 0.77042975990070553534903399964, 2.17257943216708447297551855613, 4.72023700573963724994095112963, 5.29219674103519486875835112806, 5.77204909162327596713233895424, 7.04773112189335544184998369581, 7.934565546319707120258696666171, 8.892941861234655334363081072810, 9.922619051600018879076128205456, 10.55410614714964632169039307744

Graph of the $Z$-function along the critical line